On a no carb diet the body breaks down protein

If Protein lost on a no carb diet can not be used to build muscle then a diet with the same amount of protein but all Carbohydrate Calories replaced with Fat Calories would increase loss or decrease gains of muscle

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No, the body cannot convert fat directly into carbohydrates without the involvement of protein. The primary way to produce glucose is through gluconeogenesis, which mainly uses amino acids derived from proteins.